Applied Electronics II · 120 h · Topic 4 of 8

Frequency response of feedback amplifiers

The same amplifier that is stable with a gain of 100 oscillates as a follower. The lower the feedback gain, the harder it is to stabilize.

Gain–bandwidth Complex poles Quality factor Phase margin Overshoot

01One pole: the gain–bandwidth product

If the amplifier has a single pole, \( A(s) = A_0/(1 + s/\omega_1) \), feedback shifts it. Substituting into \( A/(1 + A\beta) \):

\[ A_f(s) = \frac{A_0/(1+T_0)}{1 + s/\big[\omega_1 (1 + T_0)\big]} \] \( T_0 = A_0\beta \) is the DC loop gain. The gain goes down by \( 1 + T_0 \) and the pole goes up by the same factor: the gain–bandwidth product is maintained. A 1 MHz op amp gives 100 kHz at a gain of 10 and 10 kHz at a gain of 100.

With a single pole, the feedback system is never unstable: the pole moves along the real axis to the left, the farther the greater the loop gain. The step response is an exponential with no overshoot. The poles of the amplifier without feedback were calculated in frequency response of amplifiers.

02Two poles: the quality factor

With two real poles \( \omega_1 \) and \( \omega_2 \), as the loop gain increases the feedback poles approach each other along the real axis, meet and separate as a complex pair. The system remains stable, but a peak appears in the frequency response and an overshoot in the step response:

\[ \omega_0 = \sqrt{(1 + T_0)\,\omega_1 \omega_2} \] \[ Q = \frac{\sqrt{(1 + T_0)\,\omega_1 \omega_2}}{\omega_1 + \omega_2} \] With Q = 0.5 the poles coincide: it is the fastest response without overshoot. With Q = 0.707 the frequency response is maximally flat, Butterworth, and the step overshoots by 4.3 %. Above that, the peak grows with Q.

03Three poles: instability

With three or more poles, beyond a certain loop gain the complex pair crosses into the right half-plane and the amplifier oscillates. The practical way to assess it is to look at the loop gain \( T(j\omega) \): at the crossover frequency, where \( |T| = 1 \), the phase has to be far from −180°. That distance is the phase margin, and it sets the shape of the feedback response:

Phase margin	Q of the dominant pair	Frequency peaking	Step overshoot
76°	0.5	None	None
65.5°	0.707	None (Butterworth)	4.3 %
60°	0.82	0.3 dB	8.8 %
45°	1.19	2.4 dB	23 %

The values in the table are exact for a loop with a dominant pole and a second pole, and a good guide in the other cases. The usual design asks for at least 45°, and preferably 60°.

There is a surprising consequence: an amplifier is harder to stabilize the lower the feedback gain. With large β, \( |T| \) is larger at all frequencies, the crossover moves up toward where the higher poles act, and the phase margin shrinks. The most demanding case is the voltage follower, with β = 1.

Lab · poles, phase margin and peaking

An amplifier with gain \( 10^5 \) and poles at 1 kHz, 1 MHz and 10 MHz, fed back with a resistive β. The gray curve is the gain without feedback; the dotted yellow one, \( 1/\beta \), and the blue one, the feedback gain. The crossing of the gray with the yellow is the frequency where \( |T| = 1 \).

Ideal feedback gain 1/β =

The remedy, compensation, modifies the gain without feedback so that the crossover occurs where the phase is still comfortable. It is the subject of the next unit.

04In the lab

Activity 1 · Gain–bandwidth product

Measure the bandwidth of a noninverting amplifier with a compensated op amp for gains of 1, 10 and 100, and verify that the product is maintained.

Activity 2 · Overshoot

Apply a small step to a follower with an increasing load capacitor and measure the overshoot. Estimate the phase margin with the table.

Activity 3 · Loop gain

Measure the loop gain of a feedback amplifier by injecting a signal in series with the loop, using the Middlebrook method, and read the phase margin.

05Common mistakes

Believing that more feedback always stabilizes. With three poles, more loop gain brings the circuit closer to oscillation.
Looking at the phase of the feedback gain. Stability is judged with the loop gain \( T \), not with \( A_f \).
Forgetting the load capacitance. It adds a pole with the output resistance and can make a stable follower oscillate.
Confusing bandwidth with crossover frequency. They coincide only with large phase margins.

06Self-assessment

An amplifier with gain 10⁴ and a pole at 100 Hz is fed back with β = 0.01. Where does the pole end up?

At \( 100 \cdot 101 = 10.1 \) kHz, with gain 99.

What Q corresponds to a maximally flat response?

\( Q = 1/\sqrt 2 \approx 0.707 \).

An amplifier has 30° of phase margin as a follower. Does it improve or worsen with a gain of 10?

It improves: the loop gain goes down, the crossover drops in frequency and the phase is less negative there.

What overshoot will an amplifier with 60° of phase margin have?

About 9 %.

07Further reading

Paul R. Gray, Paul J. Hurst, Stephen H. Lewis and Robert G. Meyer. Analysis and Design of Analog Integrated Circuits. 5th ed., Wiley, 2009. Frequency response and stability of feedback amplifiers.
Sergio Franco. Design with Operational Amplifiers and Analog Integrated Circuits. 4th ed., McGraw-Hill, 2015. Loop gain, phase margin and their measurement in op amp circuits.
Adel S. Sedra and Kenneth C. Smith. Microelectronic Circuits. 8th ed., Oxford University Press, 2020. The movement of the poles with feedback, with two-pole and three-pole examples.

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Stability, compensation, oscillators and slew rate